Is it permissible to deliver sermons and speeches on mixed-gender public transportation if that might lead to quarrels?

The basic principle is that the caller to Allah should not be prevented from calling to Allah in any place or time that is feasible, while observing wisdom in the call, and choosing the time, subject, and method. This is in emulation of the Prophet, peace and blessings be upon him, and his companions, who would alternate their admonitions to people out of fear of their becoming weary and bored, and be moderate in their preaching so that hearts would not tire of it. Admonishing in mixed transportation is acceptable in principle, but the quantity, quality, time, and situation should be considered. The best of affairs are their middles.